3.307 \(\int \frac{\cos (x)}{a+b \sin ^2(x)} \, dx\)

Optimal. Leaf size=25 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} \sin (x)}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b}} \]

[Out]

ArcTan[(Sqrt[b]*Sin[x])/Sqrt[a]]/(Sqrt[a]*Sqrt[b])

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Rubi [A]  time = 0.0288314, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {3190, 205} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} \sin (x)}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b}} \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]/(a + b*Sin[x]^2),x]

[Out]

ArcTan[(Sqrt[b]*Sin[x])/Sqrt[a]]/(Sqrt[a]*Sqrt[b])

Rule 3190

Int[cos[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Sin[e + f*x], x]}, Dist[ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*ff^2*x^2)^p, x], x, Sin[e +
f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{\cos (x)}{a+b \sin ^2(x)} \, dx &=\operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\sin (x)\right )\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt{b} \sin (x)}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b}}\\ \end{align*}

Mathematica [A]  time = 0.0074584, size = 25, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} \sin (x)}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b}} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]/(a + b*Sin[x]^2),x]

[Out]

ArcTan[(Sqrt[b]*Sin[x])/Sqrt[a]]/(Sqrt[a]*Sqrt[b])

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Maple [A]  time = 0.027, size = 17, normalized size = 0.7 \begin{align*}{\arctan \left ({\sin \left ( x \right ) b{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)/(a+b*sin(x)^2),x)

[Out]

1/(a*b)^(1/2)*arctan(sin(x)*b/(a*b)^(1/2))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)/(a+b*sin(x)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.03186, size = 190, normalized size = 7.6 \begin{align*} \left [-\frac{\sqrt{-a b} \log \left (-\frac{b \cos \left (x\right )^{2} + 2 \, \sqrt{-a b} \sin \left (x\right ) + a - b}{b \cos \left (x\right )^{2} - a - b}\right )}{2 \, a b}, \frac{\sqrt{a b} \arctan \left (\frac{\sqrt{a b} \sin \left (x\right )}{a}\right )}{a b}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)/(a+b*sin(x)^2),x, algorithm="fricas")

[Out]

[-1/2*sqrt(-a*b)*log(-(b*cos(x)^2 + 2*sqrt(-a*b)*sin(x) + a - b)/(b*cos(x)^2 - a - b))/(a*b), sqrt(a*b)*arctan
(sqrt(a*b)*sin(x)/a)/(a*b)]

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Sympy [A]  time = 1.63744, size = 87, normalized size = 3.48 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{\sin{\left (x \right )}} & \text{for}\: a = 0 \wedge b = 0 \\\frac{\sin{\left (x \right )}}{a} & \text{for}\: b = 0 \\- \frac{1}{b \sin{\left (x \right )}} & \text{for}\: a = 0 \\- \frac{i \log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + \sin{\left (x \right )} \right )}}{2 \sqrt{a} b \sqrt{\frac{1}{b}}} + \frac{i \log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} + \sin{\left (x \right )} \right )}}{2 \sqrt{a} b \sqrt{\frac{1}{b}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)/(a+b*sin(x)**2),x)

[Out]

Piecewise((zoo/sin(x), Eq(a, 0) & Eq(b, 0)), (sin(x)/a, Eq(b, 0)), (-1/(b*sin(x)), Eq(a, 0)), (-I*log(-I*sqrt(
a)*sqrt(1/b) + sin(x))/(2*sqrt(a)*b*sqrt(1/b)) + I*log(I*sqrt(a)*sqrt(1/b) + sin(x))/(2*sqrt(a)*b*sqrt(1/b)),
True))

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Giac [A]  time = 1.121, size = 22, normalized size = 0.88 \begin{align*} \frac{\arctan \left (\frac{b \sin \left (x\right )}{\sqrt{a b}}\right )}{\sqrt{a b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)/(a+b*sin(x)^2),x, algorithm="giac")

[Out]

arctan(b*sin(x)/sqrt(a*b))/sqrt(a*b)